Q. 47 - Precalculus Enhanced with Graphing Utilities

Question

Determine the quadratic function whose graph is given.

Step-by-Step Solution

Answer

The quadratic function whose graph is given is $f\left(x\right)=(x+1{)}^2-2$ or $f\left(x\right)=x^2+2x-1$.

Step 1. Given information.

The given graph is:

Step 2. Determine the equation.

The vertex is $(-1,-2)$, so $h=-1$ and $k=-2$. Substitute these values into equation $f\left(x\right)=a(x-h{)}^2+k$.

$$\begin{gathered} f\left(x\right)=a(x-(-1){)}^2+(-2\left) \\ f\right(x)=a(x+1{)}^2-2 \end{gathered}$$

To determine the value of $a$, we use the fact that $f\left(0\right)=-1$ (the y-intercept).

$\begin{array}{rcl}-1& =& a(0+1{)}^2-2\\ -1+2& =& a\\ 1& =& a\end{array}$

Substitute $a=1$ in $f\left(x\right)=a(x+1{)}^2-2$.

$$\begin{gathered} f\left(x\right)=1(x+1{)}^2-2 \\ f(x)=(x+1{)}^2-2 \\ f\left(x\right)=x^2+2x+1-2 \\ f\left(x\right)=x^2+2x-1 \end{gathered}$$

Step 3. Conclusion.

The quadratic function whose graph is given is $f\left(x\right)=(x+1{)}^2-2$ or $f\left(x\right)=x^2+2x-1$.